On sums of dependent random lifetimes under the Time Transformed Exponential model

TL;DR

This paper derives analytical formulas for the sum of two dependent lifetimes modeled by a Time Transformed Exponential distribution, using a novel multivariate distortion approach that simplifies prediction and analysis.

Contribution

It introduces an alternative to copula-based methods by employing multivariate distortions for the TTE model, enabling easier computation of sums and conditional distributions.

Findings

Derived explicit distribution formulas for sums of dependent lifetimes.

Demonstrated how to predict sums using quantile regression techniques.

Provided a new analytical framework for dependent lifetime analysis.

Abstract

Considered a pair of random lifetimes whose dependence is described by a Time Transformed Exponential model, we provide analytical expressions for the distribution of their sum. These expressions are obtained by using a representation of the joint distribution in terms of multivariate distortions, which is an alternative approach to the classical copula representation. Since this approach allows to obtain conditional distributions and their inverses in simple form, then it is also shown how it can be used to predict the value of the sum from the value of one of the variables (or vice versa) by using quantile regression techniques.